associatedCastelnuovoSurface -- Castelnuovo surface associated to a rational complete intersection of three quadrics in P^7

• Usage:

  U' = associatedCastelnuovoSurface X

• Inputs:
  • X, a complete intersection of three quadrics in PP^7, containing a surface $S\subset Y$ that admits a congruence of $(2e-1)$-secant curves of degree $e$ inside the ambient fivefold $Y$ of the fourfold $X$
• Optional inputs:
  • Singular => ..., default value null, whether to transfer computation to Singular
  • Strategy => ..., default value null
  • Verbose => ..., default value false, request verbose feedback
• Outputs:
  • U', the (minimal) Castelnuovo surface associated to $X$
• Consequences:

  • building U' will return the following four objects:
    • the dominant rational map $\mu:Y\dashrightarrow W$ defined by the linear system of hypersurfaces of degree $2e-1$ having points of multiplicity $e$ along $S$;
    • the surface $U\subset W$ determining the inverse map of the restriction of $\mu$ to $X$;
    • the list of the exceptional curves on the surface $U$;
    • a rational map of degree 1 from the surface $U$ to the minimal Castelnuovo surface $U'$.

i1 : X = specialFourfold random({5:{1}},0_(PP_(ZZ/33331)^7));

o1 : ProjectiveVariety, complete intersection of three quadrics in PP^7 containing a surface of degree 1 and sectional genus 0

i2 : describe X

o2 = Complete intersection of 3 quadrics in PP^7
     of discriminant 31 = det| 8 1 |
                             | 1 4 |
     containing a surface of degree 1 and sectional genus 0
     cut out by 5 hypersurfaces of degree 1
     (This is a classical example of rational fourfold)

i3 : time U' = associatedCastelnuovoSurface X;
 -- used 2.97927s (cpu); 1.26717s (thread); 0s (gc)

o3 : ProjectiveVariety, Castelnuovo surface associated to X

i4 : (mu,U,C,f) = building U';

i5 : ? mu

o5 = multi-rational map consisting of one single rational map
     source variety: 5-dimensional subvariety of PP^7 cut out by 2 hypersurfaces of degree 2
     target variety: PP^4
     dominance: true